The tricritical Ising CFT and conformal bootstrap

TL;DR

This work investigates the tricritical Ising CFT as the IR fixed point of λφ^6 theory and its interpolation between the perturbative ε-expansion near the upper critical dimension $d=3$ and the exact 2d minimal model. By performing mixed-correlator conformal bootstrap with external operators {φ, φ^2, φ^3}, the authors locate bootstrap islands in $d=2.75$ and $d=2.5$ that are consistent with the 3d perturbative predictions and the 2d exact values, while no islands emerge at $d=2$ and $d=2.25$ under the current setup. The paper also surveys perturbative spectra, introduces one-loop dilatation operator results, and discusses the bootstrap methodology, Navigator searches, and extremal-spectrum analyses. These findings support a scenario where diagonal minimal models $ ext{M}_{k+2,k+1}$ connect to φ^{2k} theories via a continuous family of CFTs parameterized by $d$, and lay groundwork for exploring tricritical O(n) and multicritical models with bootstrap methods.

Abstract

The tricritical Ising CFT is the IR fixed-point of $λφ^6$ theory. It can be seen as a one-parameter family of CFTs connecting between an $\varepsilon$-expansion near the upper critical dimension 3 and the exactly solved minimal model in $d=2$. We review what is known about the tricritical Ising CFT, and study it with the numerical conformal bootstrap for various dimensions. Using a mixed system with three external operators $\{φ\simσ,φ^2\sim ε,φ^3\simσ'\}$, we find three-dimensional "bootstrap islands" in $d=2.75$ and $d=2.5$ dimensions consistent with interpolations between the perturbative estimates and the 2d exact values. In $d=2$ and $d=2.25$ the setup is not strong enough to isolate the theory. This paper also contains a survey of the perturbative spectrum and a review of results from the literature.

Figures (19)

• Figure 1: Continuous families of CFTs discussed in this work. The circles represent unitary theories in integer dimensions. The vertical axis is a proxy for interaction strength, and $c$ is the 2d central charge.
• Figure 2: Figure describing our setup and results. The error-bars in $d=2.5$ and $d=2.75$ represent our three-dimensional islands \ref{['eq:island275']}--\ref{['eq:island25']}. Solid lines are Padé approximants \ref{['eq:Pades']} and the dashed lines are the spectral gaps we assume above the external operators in the ${\mathbb Z}_2$-even (blue tones) and ${\mathbb Z}_2$-odd (orange tones) sectors. The islands are almost invisible at this scale, see figure \ref{['fig:islandsLog']} for a log plot of anomalous dimensions. In $d=2$ and $d=2.25$ we do not find islands with the current precision.
• Figure 3: Logarithmic plot of our islands \ref{['eq:island275']}--\ref{['eq:island25']} in terms of anomalous dimensions $\gamma$. The three intervals for a fixed $d$ represent three-dimensional islands (rigorous intervals) and the solid lines are Padé approximants \ref{['eq:Pades']}.
• Figure 4: Navigator run in $d=2.75$ projected onto the subspace $(\Delta_\phi,\Delta_{\phi^2})$. Gray is positive navigator values (ruled-out points), purple is negative (allowed points). We display plots including the $\Delta_{\phi^3}$ axis in figure \ref{['fig:nav-275-more']}. The rectangle represents our rigorous bound, found using additional navigator searches along the different axes (appendix \ref{['app:NavigatorIslands']}), and the red circle the Padé approximant \ref{['eq:Pades']}. Although the starting point (dark star) is farther away from the allowed points than the Padé approximant, the navigator search still finds the island.
• Figure 5: Schematic phase diagram of the Blume--Capel model \ref{['eq:BCmodel']}. Along the dashed line, the phase transition is first-order; along the solid line, the phase transition is second-order and described by the Ising CFT. The first- and second-order transitions meet at the tricritical point. Figure adapted from Cardy:1996xt.